Artificial Intelligence SS 2017

1. Artificial IntelligenceSS 2017 Search Methods Anna Fensel 27.03.2017

2. Next Lecture

3. Outline • Motivation • Technical Solution • Uninformed Search • Depth-First Search • Breadth-First Search • Informed Search • Best-First Search • Hill Climbing • A* • Illustration by a Larger Example • Extensions • Summary

4. Problem Solving as Search MOTIVATION

5. Motivation • One of the major goals of AI is to help humans in solving complex tasks • How can I fill my container with pallets? • Which is the shortest way from Milan to Innsbruck? • Which is the fastest way from Milan to Innsbruck? • How can I optimize the load of my freight to maximize my revenue? • How can I solve my Sudoku game? • What is the sequence of actions I should apply to win a game? • Sometimes finding a solution is not enough, you want the optimal solution according to some “cost” criteria • All the example presented above involve looking for a plan • A plan that can be defined as the set of operations to be performed of an initial state, to reach a final state that is considered the goal state • Thus we need efficient techniques to search for paths, or sequences of actions, that can enable us to reach the goal state, i.e. to find a plan • Such techniques are commonly called Search Methods

6. Examples of Problems: Towers of Hanoi • 3 pegs A, B, C • 3 discs represented as natural numbers (1, 2, 3) which correspond to the size of the discs • The three discs can be arbitrarily distributed over the three pegs, such that the following constraint holds: di is on top of dj → di < dj • Initial status: ((123)()()) • Goal status: (()()(123)) 1 2 3 Operators: Move disk to peg Applying: Move 1 to C (1 → C) to the initial state ((123)()()) a new state is reached ((23)()(1)) Cycles may appear in the solution! A B C

7. Examples of Problems: Blocksworld E D C B E A B C A D • Objects: blocks • Attributes (1-ary relations): cleartop(x), ontable(x) • Relations: on(x,y) • Operators: puttable(x) where x must be cleartop; put(x,y), where x and y must be cleartop Initial State Goal State • Initial state: • ontable(E), cleartop(E) • ontable(A), cleartop(A) • ontable(B), cleartop(B) • ontable(C) • on(D,C), cleartop (D) • Applying the move put(E,A): • on(E,A), cleartop(E) • ontable(A) • ontable(B), cleartop(B) • ontable(C) • on(D,C), cleartop (D)

8. Search Methods TECHNICAL SOLUTION

9. Search Space Representation Node • Representing the search space is the first step to enable the problem resolution • Search space is mostly represented through graphs • A graph is a finite set of nodes that are connected by arcs • A loop may exist in a graph, where an arc lead back to the original node • In general, such a graph is not explicitly given • Search space is constructed during search Loop Arc

10. Search Space Representation undirect direct • A graph is undirected if arcs do not imply a direction, direct otherwise • A graph is connected if every pair of nodes is connected by a path • A connected graph with no loop is called tree • A weighted graph, is a graph for which a value is associated to each arc connected disconnected tree weighted 1 1 4 5 6 2 1 2 1

11. Example: Towers of Hanoi* These nodesare equals 2 2 1 1 1 3 3 A B C A B C 2 3 … A B C 2 1 1 3 2 3 2 3 1 A B C A B C A B C … 1 1 1 1 3 2 3 2 3 2 3 2 A B C A B C A B C A B C … … … … … … 1 2 3 A B C … … * A partial tree search space representation

12. Example: Towers of Hanoi* * A complete direct graph representation[http://en.wikipedia.org/wiki/Tower_of_Hanoi]

13. Search Methods • A search method is defined by picking the order of node expansion • Strategies are evaluated along the following dimensions: • completeness: does it always find a solution if one exists? • time complexity: number of nodes generated • space complexity: maximum number of nodes in memory • optimality: does it always find the shortest path solution? • Time and space complexity are measured in terms of • b: maximum branching factor of the search tree • d: depth of the shortest path solution • m: maximum depth of the state space (may be ∞)

14. Search Methods • Uninformed techniques • Systematically search complete graph, unguided • Also known as brute force, naïve, or blind • Informed methods • Use problem specific information to guide search in promising directions

15. Brute force approach to explore search space UNINFORMED SEARCH

16. Uninformed Search • A class of general purpose algorithms that operates in a brute force way • The search space is explored without leveraging on any information on the problem • Also called blind search, or naïve search • Since the methods are generic they are intrinsically inefficient • E.g. Random Search • This method selects randomly a new state from the current one • If the goal state is reached, the search terminates • Otherwise the methods randomly select an other operator to move to the next state • Prominent methods: • Depth-First Search • Breadth-First Search • Uniform-Cost Search

17. Depth-First Search • Depth-First Search (DFS) begins at the root node and exhaustively search each branch to its maximum depth till a solution is found • The successor node is selected going in depth using from right to left (w.r.t. graph representing the search space) • If greatest depth is reached with no solution, we backtrack till we find an unexplored branch to follow • DFS is not complete • If cycles are presented in the graph, DFS will follow these cycles indefinitively • If there are no cycles, the algorithm is complete • Cycles effects can be limited by imposing a maximal depth of search (still the algorithm is incomplete) • DFS is not optimal • The first solution is found and not the shortest path to a solution • The algorithm can be implemented with a Last In First Out (LIFO) stack or recursion

18. Depth-First Search: Algorithm List open, closed, successors={}; Node root_node, current_node; insert-first(root_node,open) while not-empty(open); current_node= remove-first(open); insert-first (current_node,closed); if (goal(current_node)) return current_node; else successors=successorsOf(current_node); for(x in successors) if(not-in(x,closed)) insert-first(x,open); endIf endWhile N.B.= this version is not saving the path for simplicity

19. Depth-First Search: Example S 1 A B 2 3 C S B A D 5 open={S} closed ={} 0. Visit S: open={A,B}, closed={S} • Visit A: open={S,B,F,B}, closed={A,S} • Visit S: open={B,F,B}, closed={S,A,S} • Visit B: open={S,A,F,D,F,B}, closed={B,S,A,S} • Visit S: open={A,F,D,F,B}, closed={S,B,S,A,S} • Visit A: open={F,D,F,B}, closed={A,S,B,S,A,S} • Visit F: GOAL Reached! 4 6 S A D F F F

20. Depth-First Search: Example S A B C S B A D S A D F F F Result is: S->A->B->F

21. Depth-First Search: Complexity d=0 • Time Complexity • assume (worst case) that there is 1 goal leaf at the RHS • so DFS will expand all nodes =1 + b + b2+ ......... + bm = O (bm) • where m is the max depth of the tree • Space Complexity • how many nodes can be in the queue (worst-case)? • at each depth l < d we have b-1 nodes • at depth m we have b nodes • total = (d-1)*(b-1) + b = O(bm) d=1 m=d=2 G d=0 d=1 d=2 d=3 m=d=4

22. Breadth-First Search • Breadth-First Search (BFS) begins at the root node and explore level-wise al the branches • BFS is complete • If there is a solution, BFS will found it • BFS is optimal • The solution found is guaranteed to be the shortest path possible • The algorithm can be implemented with a First In First Out (FIFO) stack

23. Breadth-First Search: Algorithm List open, closed, successors={}; Node root_node, current_node; insert-last(root_node,open) while not-empty(open); current_node=remove-first(open); insert-last(current_node,closed); if (goal(current_node)) return current_node; else successors=successorsOf(current_node); for(x in successors) if(not-in(x,closed))insert-last(x,open); endIf endWhile N.B.= this version is not saving the path for simplicity

24. Breadth-First Search: Example S 1 A 2 B 3 4 S B C 5 A D open = {S}, closed={} 0. Visit S: open = {A,B}, closed={S} • Visit A: open={B,S,B,F}, closed={S,A} • Visit B: open={S,B,F,F,A,C,D}, closed={S,A,B} • Visit S: open={B,F,F,A,C,D}, closed={S,A,B,S} • Visit B: open={F,F,A,C,D,S,A,C,D}, closed={S,A,B,S,B} • Visit F: Goal Found! S A D F F F

25. Breadth-First Search: Example S A B C S B A D S A D F F F Result is: S->A->F

26. Breadth-First Search: Complexity • Time complexity is the same magnitude as DFS • O (bm) • where m is the depth of the solution • Space Complexity • how many nodes can be in the queue (worst-case)? • assume (worst case) that there is 1 goal leaf at the RHS • so BFS will store all nodes =1 + b + b2+ ......... + bm = O (bm) d=0 1 d=1 2 3 d=2 d=3 d=4 7 4 6 5 G 8 9 10 11 12 13 14 15

27. Further Uninformed Search Strategies • Depth-limited search (DLS): Impose a cut-off (e.g. n for searching a path of length n-1), expand nodes with max. depth first until cut-off depth is reached (LIFO strategy, since it is a variation of depth-first search). • Bidirectional search (BIDI): forward search from initial state & backward search from goal state, stop when the two searches meet. Average effort O(bd/2) if testing whether the search fronts intersect has constant effort • In AI, the problem graph is typically not known. If the graph is known, to find all optimal paths in a graph with labelled arcs, standard graph algorithms can be used

28. Using knowledge on the search space to reduce search costs INFORMED SEARCH

29. Informed Search • Blind search methods take O(bm) in the worst case • May make blind search algorithms prohibitively slow where d is large • How can we reduce the running time? • Use problem-specific knowledge to pick which states are better candidates

30. Informed Search • Also called heuristic search • In a heuristic search each state is assigned a “heuristic value” (h-value) that the search uses in selecting the “best” next step • A heuristic is an operationally-effective nugget of information on how to direct search in a problem space • Heuristics are only approximately correct

31. Informed Search: Prominent methods • Best-First Search • A* • Hill Climbing

32. Cost and Cost Estimation f(n)=g(n)+h(n) • g(n) the cost (so far) to reach the node n • h(n) estimated cost to get from the node to the goal • f(n) estimated total cost of path through n to goal

33. Informed Search: Best-First Search • Special case of breadth-first search • Uses h(n) = heuristic function as its evaluation function • Ignores cost so far to get to that node(g(n)) • Expand the node that appears closest to goal • Best First Search is complete • Best First Search is not optimal • A solution can be found in a longer path (higher h(n) with a lower g(n) value) • Special cases: • uniform cost search: f(n) = g(n) = path to n • A* search 33

34. Best-First Search: Algorithm List open, closed, successors={}; Node root_node, current_node; insert-last(root_node,open) while not-empty(open); current_node=remove-first (open); insert-last(current_node,closed); if (goal(current_node)) return current_node; else successors=estimationOrderedSuccessorsOf(current_node); for(x in successors) if(not-in(x,closed)) insert-last(x,open); endIf endWhile returns the list of direct descendants of the current node in shortest cost order N.B.= this version is not saving the path for simplicity 34

35. Best-First Search: Example S 1 h=1 h=1 A 2 B h=2 h=2 h=2 h=2 3 h=2 h=2 h=2 S B C A D open = {S}, closed={} 0. Visit S: open = {A,B}, closed={S} • Visit A: open={B,F,B,S}, closed={S,A} • Visit B: open={F,B,S,F,A,C,D}, closed={S,A,B} • Visit F: Goal Found! S A D F F F In this case we estimate the cost as the distance from the root node (in term of nodes) 35

36. Best-First Search: Example S h=1, w=2 h=1, w=1 A B h=2, w=7 h=2, w=4 h=2 h=2 h=2 h=2 h=2 S B C A D S A D F F F Result is: S->A->F! If we consider real costs, optimal solution is: S->B->F 36

37. A* • Derived from Best-First Search • Uses both g(n) and h(n) • A* is optimal • A* is complete

38. A* : Algorithm List open, closed, successors={}; Node root_node, current_node, goal; insert-back(root_node,open) while not-empty(open); current_node=remove-front(open); insert-back(current_node,closed); if (current_node==goal) return current_node; else successors=totalEstOrderedSuccessorsOf(current_node); for(x in successors) if(not-in(x,closed)) insert-back(x,open); endIf endWhile returns the list of direct descendants of the current node in shortest total estimation order N.B.= this version is not saving the path for simplicity 38

39. A* : Example S 1 h=1, w=2, g=2 h=1, w=1, g=1 A 2 B h=2, w=7, g=9 h=2, w=4, g=5 h=2, w=2, g=4 h=2, w=1 g=3 h=2, w=4, g=5 3 h=2 w=3 g=4 h=2,w=1 g=2 S B C A D 5 4 open = {S}, closed={} 0. Visit S: open = {B,A}, closed={S} • Visit B: open={A,C,A,F,D}, closed={S,B} • Visit A: open={C,A,F,D,B,S,F}, closed={S,B,A} • Visit C: open={A,F,D,B,S,F}, closed={S,B,A,C} • Visit A: open={F,D,B,S,F}, closed={S,B,A,C,A} • Visit F: Goal Found! S A D F F F In this case we estimate the cost as the distance from the root node (in term of nodes) 39

40. A* : Example S h=1, w=2, g=2 h=1, w=1, g=1 A B h=2, w=7, g=9 h=2, w=4, g=5 h=2, w=2, g=4 h=2, w=1 g=3 h=2, w=4, g=5 h=2 w=3 g=4 h=2,w=1 g=2 S B C A D S A D F F F Result is: S->B->F! 40

41. Hill Climbing • Special case of depth-first search • Uses h(n) = heuristic function as its evaluation function • Ignores cost so far to get to that node(g(n)) • Expand the node that appears closest to goal • Hill Climbing is not complete • Unless we introduce backtracking • Hill Climbing is not optimal • Solution found is a local optimum

42. Hill Climbing: Algorithm List successors={}; Node root_node, current_node, nextNode; current_node=root_node while (current_node!=null) if (goal(current_node)) return current_node; else successors=successorsOf(current_node); nextEval = -∞; nextNode=null; for(x in successors) if(eval(x)> nextEval) nexEval=eval(x); nextNode=x; current_node=nextNode, endIf endWhile N.B.= this version is not using backtracking 42

43. Hill Climbing: Example S 1 h=1 h=1 A B h=2 h=2 h=2 h=2 h=2 2 h=3 h=2 S B C A D h=1 0. current_node=S, successors (A=1,B=1) • current_node=A, successors (B=2,F=2,S=2) • current_node=B, successors (F=1,….) • current_node=F: Goal Found! 3 S A D F F F In this case we estimate the cost as the distance from the root node (in term of nodes) 43

44. Hill Climbing: Example S h=1 h=1 A B h=2 h=2 h=2 h=2 h=2 h=3 h=2 S B C A D h=1 S A D F F F Result is: S->A->B->F! Not optimal, more if at step 1 h(S)=2 we would have completed without funding a solution 44

45. Informed Search Algorithm Comparison b, branching factor d, tree depth of the solution m, maximum tree depth 45

46. ILLUSTRATION BY A LARGER EXAMPLE

47. Route Search • Start point: Milan • End point: Innsbruck • Search space: Cities • Nodes: Cities • Arcs: Roads • Let’s find a possible route!

48. Graph Representation • We start from the root node, and pick the leaves • The same apply to each leaves • But we do not reconsider already used arcs • The first node picked is the first node on the right Feldkirck Innsbruck (Goal) Landeck Merano Bolzano Chiavenna Sondrio Trento Lugano 20 Lecco Como Bergamo Milan (Root) Verona Brescia Piacenza

49. Depth-First Search Milan 1 Piacenza Bergamo Como Lecco 2 Brescia Brescia Trento Verona Trento Verona 3 Bolzano Trento Bolzano Trento 4 Bolzano Bolzano Merano 5 Merano Innsbruck Innsbruck Merano According to Google Maps: 464 km – 4 hours 37 mins Merano Landeck Innsbruck Landeck Innsbruck Landeck Landeck Innsbruck Innsbruck Innsbruck Innsbruck N.B.: by building the tree, we are exploring the search space!

50. Breadth-First search Milan 1 4 2 3 Piacenza Bergamo Como Lecco 8 5 7 6 9 10 Brescia Brescia Lecco Sondrio Chiavenna Lugano 13 12 14 21 11 15 20 18 19 17 16 Trento Verona Trento Verona Lugano Merano Sondrio Landeck Chi. Chi. Feld. 25 23 24 22 26 Bolzano Trento Bolzano Trento Innsbruck Bolzano Bolzano Innsbruck Innsbruck Merano According to Google Maps: 358 km – 5 hours 18 mins Merano Innsbruck Innsbruck Landeck Landeck Innsbruck Innsbruck N.B.: by building the tree, we are exploring the search space!